## 8.1. Pointwise Convergence

### Example 8.1.8 (c): Pointwise Convergence does not preserve Integrability

Find a pointwise convergent sequence of Riemann-integrable functions whose
limit is not Riemann-integrable.

Let *r _{n}* be the (countable) set of rational numbers
inside the interval

*[0, 1]*, ordered in some way, and define the functions

and

Then:

- The sequence
gconverges pointwise to_{n}g- Each
ghas only finitely many points of discontinuity, thus each_{n}gis Riemann integrable_{n}- The limit function
gis not Riemann integrable

Thus, pointwise convergence does not preserve Riemann-integrability.